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Be careful while driving here. The police seem to stop cars quite often. If you are pulled over, the driver will be asked for a passport and license, and possibly for all passenger passports. Most officers are legitimate and courteous, but some are less than scrupulous and may ask for a bribe, threatening to hold your passport if you do not pay.
Although it may be impossible to avoid, try not give any officer a reason to stop you (e.g., don't speed) or else they may shake you down.
Be aware of routine checkpoint on the roads. The photocopy of the inside page of the passport may not be enough, especially if the police officer wants to verify date of entry into the country. Perhaps you can use the rental car agreement, showing when the car was picked up at the airport.
You may also be stopped just walking around Panama City. Definitely carry your passport always and be ready to deal with corrupt policemen in Panama!